Best Known (186−33, 186, s)-Nets in Base 2
(186−33, 186, 320)-Net over F2 — Constructive and digital
Digital (153, 186, 320)-net over F2, using
- 21 times duplication [i] based on digital (152, 185, 320)-net over F2, using
- t-expansion [i] based on digital (151, 185, 320)-net over F2, using
- trace code for nets [i] based on digital (3, 37, 64)-net over F32, using
- net from sequence [i] based on digital (3, 63)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 3 and N(F) ≥ 64, using
- net from sequence [i] based on digital (3, 63)-sequence over F32, using
- trace code for nets [i] based on digital (3, 37, 64)-net over F32, using
- t-expansion [i] based on digital (151, 185, 320)-net over F2, using
(186−33, 186, 803)-Net over F2 — Digital
Digital (153, 186, 803)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2186, 803, F2, 2, 33) (dual of [(803, 2), 1420, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2186, 1040, F2, 2, 33) (dual of [(1040, 2), 1894, 34]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2184, 1039, F2, 2, 33) (dual of [(1039, 2), 1894, 34]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2184, 2078, F2, 33) (dual of [2078, 1894, 34]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2183, 2077, F2, 33) (dual of [2077, 1894, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,14]) [i] based on
- linear OA(2177, 2049, F2, 33) (dual of [2049, 1872, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 2049 | 222−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(2155, 2049, F2, 29) (dual of [2049, 1894, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 2049 | 222−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(26, 28, F2, 3) (dual of [28, 22, 4]-code or 28-cap in PG(5,2)), using
- discarding factors / shortening the dual code based on linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to C([0,16]) ⊂ C([0,14]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2183, 2077, F2, 33) (dual of [2077, 1894, 34]-code), using
- OOA 2-folding [i] based on linear OA(2184, 2078, F2, 33) (dual of [2078, 1894, 34]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2184, 1039, F2, 2, 33) (dual of [(1039, 2), 1894, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2186, 1040, F2, 2, 33) (dual of [(1040, 2), 1894, 34]-NRT-code), using
(186−33, 186, 20544)-Net in Base 2 — Upper bound on s
There is no (153, 186, 20545)-net in base 2, because
- 1 times m-reduction [i] would yield (153, 185, 20545)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 49 047652 818519 548922 942882 333297 791054 432565 821658 503708 > 2185 [i]